Yes, it’s rational to vote in presidential elections

With only a year to the next election, and with the publicity starting up already, now is a good time to ask, is it rational for you to vote? And, by extension, is it worth your while to pay attention to what Hillary, Rudy, and all the others will be saying for the next year or so? With a chance of casting a decisive vote that is comparable to the chance of winning the lottery, what is the gain from being a good citizen and casting your vote?

The short answer is, quite a lot. First the bad news. With 100 million voters, your chance that your vote will be decisive–even if the national election is predicted to be reasonably close–is, at best, 1 in a million in a battleground state such as Ohio and less than 1 in 10 million or less in a less closely-fought state such as New York. (The calculation is based on the chance that your state’s vote will be exactly tied, along with the chance that your state’s electoral vote is necessary for one candidate or the other to win the Electoral College. Both these conditions are necessary for your vote to be decisive.) So voting doesn’t seem like such a good investment.

But here’s the good news. If your vote is decisive, it will make a difference for 300 million people. If you think your preferred candidate could bring the equivalent of a $50 improvement in the quality of life to the average American–not an implausible hope, given the size of the Federal budget and the impact of decisions in foreign policy, health, the courts, and other areas–you’re now buying a $1.5 billion lottery ticket. With this payoff, a 1 in 10 million chance of being decisive isn’t bad odds.

And many people do see it that way. Surveys show that voters choose based on who they think will do better for the country as a whole, rather than their personal betterment. Indeed, when it comes to voting, it is irrational to be selfish, but if you care how others are affected, it’s a smart calculation to cast your ballot, because the returns to voting are so high for everyone if you are decisive. Voting and vote choice (including related actions such as the decision to gather information in order to make an informed vote) are rational in large elections only to the extent that voters are not selfish.

That’s also the reason for contributing money to a candidate: Large contributions, or contributions to local elections, could conceivably be justified as providing access or the opportunity to directly influence policy. But small-dollar contributions to national elections, like voting, can be better motivated by the possibility of large social benefit than by any direct benefit to you. Such civically motivated behavior is consistent with both small and large anonymous contributions to charity.

The social benefit from voting also explains the declining response rates in opinion polls. In the 1950s, when mass opinion polling was rare, we would argue that it was more rational to respond to a survey than to vote in an election: for example, as one of 1000 respondents to a Gallup poll, there was a real chance that your response could noticeably affect the poll numbers (for example, changing a poll result from 49% to 50%). Nowadays, polls are so common that a telephone poll was done recently to estimate how often individuals are surveyed (the answer was about once per year). It is thus unlikely that a response to a single survey will have much impact.

So, yes, Virginia–and Ohio, and Florida, and Pennsylvania, and New Jersey–it is rational to vote. Utah, Wyoming, and Massachusetts: maybe it’s not worth your time. On the other hand, there’s a chance you could swing the national popular vote (which can affect the perception of a mandate) and in any case you’re likely to have close local races that can ultimately affect policies from schools to taxes to crime and punishment, so if you have any preferences there, it might very well be worth your time to cast your ballot and have a small chance of making a big difference.

P.S. No, I didn’t vote today. I don’t think any of the local elections were close. I doubt there was serious opposition to any of the candidates.

P.P.S. I was motivated to post this by this Freakonomics article which brings up the old argument that it’s irrational to vote because the probability of your vote being decisive is so low. Stephen Dubner writes, “The irony is that the typical voter is more likely to have an impact in a smaller election than in a larger one, but it’s the bigger elections that draw far more voters.” Actually, it’s not such an irony at all: bigger elections have bigger effects, thus more motivation to vote. It all becomes clear when you realize that most people vote because of what they think will be good for the country, not for their own personal benefit. It’s fine for individual people to disagree with this–maybe Dubner and Levitt vote for other reasons–but it’s an unfortunate blind spot for them to identify rationality with selfishness.

Good question. The answer is that you can care about others but not agree with their judgment. For example, if you are in the minority of Americans who currently support the Iraq war, you presumably think that ending the war now will be a bad thing for the whole country (or maybe regions of the world). It's not that you think that the war helps you and hurts the other Americans; it's that you disagree with them on the judgment of whether the war is a good idea. Similarly for issues such as taxes (good or bad for the economy), environmental regulations, etc. In a U.S. election, the two parties offer different baskets of candidates and policies, and it's perfectly legitimate to for you to believe that one of these baskets is likely to be much better than the other for the country as a whole, not just for your party's supporters.

Shouldn't individuals discount their expectation that their preferred candidate will actually do some good by their uncertainty about the candidate? Eg, I may like Hillary but I may also recognize my knowledge is imperfect and could be wrong. This only modifies the magnitude of the payoff, however, and doesn't undermine the logic of the argument.

"It all becomes clear when you realize that most people vote because of what they think will be good for the country, not for their own personal benefit."

- Is it not a common argument that even though we might think we vote depending on "what's best for the country as a whole", this is very sensitive to what is best for ourselves? You tend to buy into your own arguments for why candidate X is the best choice for the whole country.

We decieve ourselves in fantastic ways. Compare to results that find that most individuals actually do belive (or at least state) that they buy cars depending on safety, environment etc., but think that every one else only cares about status and prestige.

I haven't read the paper, but how can you control for these self-deception when making the above argument?

To follow up on my post. In the paper you write: “For a recent example, in the 2001 British Election Study (University of Essex 2002), only 25% of respondents thought of political activity as a good way to get ‘benefits for me and my family’, whereas 66% thought it a good way to obtain ‘benefits for groups that people care about like pensioners and the disabled’.

Sure. And when we do hypothetical experiments eliciting willingness to donate to environmental projects, we get really large estimates. When we do the exact same experiment, but with real money involved: estimates are often only a third to a half of the estimates based on the pure hypothetical scenario. I see it as very likely that survey evidence on voting motivation will be plagued by the same type of hypothetical bias.

I think the fact that one person votes can have an inflence on that person's acquaintances. In any given group of people if one of them votes, and is open about their vote, they may increase the odds that others in that group may vote. Thus, that one person's vote may turn into more than one vote.

I think there is still some problem in the argument. To simpify things a bit, your post is saying that A votes in order to make life better for B, and B votes in order to make life better for A (and both think they know better how to make life better for the other guy than the other guy him/herself).

1) In this case actually whether each voter thinks she knows better than the other voter about what's good for the other voter is irrelevant, because the other voter is not voting for her own sake anyway.
2) How is the election result different from if both A and B vote for the sake of themselves? And since the result is the same, the same problem that plagues voting for oneself plagues voting for others.
3) EVERYONE thinking they know better what's good for others than others themselves does not seem to be a persuasive argument for democracy. Also, having a random lottery to get a dictator seems to be a much more efficient way of running government, becaue the dictator cares for others too.

I think voting is just an expressive action, like shouting in a rock concert. Of course, I haven't read your paper and so this comment is only intended for your blog post.

there's a chance you could swing the national popular vote (which can affect the perception of a mandate)

How do you figure? Suppose my vote determines the popular vote ends with Bush beating Kerry by 1 vote instead of Kerry beating Bush by 1 vote. Either way, the story is that the popular vote was exceedingly close. In neither case is there a "perception of a mandate".

Suppose instead that the popular vote is somewhere in mandate territory. At what point does my vote change the perception? If my vote means Bush beats Kerry 62,040,611 to 59,028,443 instead of 62,040,610 to 59,028,444, does that affect the perception of a mandate?

There is no jump in the path from perception of mandate to non-perception of mandate. I see no scenario where one vote affects that perception in anything but a trivial way.

I agree that you can't always believe what people say. Talk is cheap. On the other hand, this provides _some_ information. Amid all the talk of pocketbook voting, most people say they vote for other reasons. I would say this information fits into the big picture. As many many people (including the Freakanomics guys) have pointed out, pocketbook voting is irrational. Given that this is a motivation that's irrational, and that most people say they don't do, that's telling us something, I think. It's not an open-and-shut case, but it's one more piece of information consistent with our story.

Eric,

Yes, there must be network effects which could multiply the value of deciding to vote. Good point.

Claker,

(1) I don't think that _everybody_ thinks they know better than others themselves. I just think that it's a rational reason to vote, that you have an opinion about which batch of policies on offer is best for the country (and the world). Consider the examples in my earlier comment.
(2) If everyone votes for themselves, the difference is then there's no reason for anyone to vote! We actually observe turnout rates of approximately 50%. Beyond this, why are you so sure they _should_ vote "for the sake of themselves"? This isn't why people say they vote, and it's not a particularly rational reason to vote. Who's in power can make a huge difference to the direction of the country, even if it doesn't have such a predictable effect on many individual voters.
(3) The dictator system has problems. For one thing, if the dictator is chosen at random there's no motivation for him or her to make the voters happy. The issue is not whether the dictator "cares for others"; it's whether he or she has good policies. Voting allows us all to choose among policies (admittedly, only among the available options, I'm not saying the system is perfect).
(4) I agree that voting is expressive and is a "consumption good" in that sense. I'm arguing that it also has a benefit in having a small chance in making the country a much better place.

Ubs,

The "mandate" thing wasn't my main argument, but, yes, I think it's possible it can make a difference; consider the (pre-WTC) status of George Bush as a minority vote-getter, or CLinton's status as someone who won less than half the popular vote.

Dan,

Thanks for the link. The paper is interesting. I recommend that you include in the abstract a sentence on the actual results. Suspense is fine, but a quick reader should be able to learn what happened right away.

Brillant. Have not read the paper yet, but if I am not mistaken, your blogpost alone debunks 50 years of reasoning on the subject.

I should like to add that I have seen German data according to which a large part of the respondents said they felt it was their "duty as a citizen" to vote. Would you concede that this is another influential factor?

I do feel our paper makes a contribution. Credit goes to Aaron (our coauthor) for coming up with the idea, but Noah and I did come up with many of the political science examples and I worked out the mathematical feedback model. Now if we can just get the Freakanomics guys to read it! I sent them a note about it once (when their earlier article on the topic came out) but I don't think they actually looked at the article; they're too busy, maybe.

I haven't seen your German data, but I believe what you're saying. After all, civic duty is one of my own motivations for voting (even though I couldn't bring myself to vote this past Tuesday, that was too much of a joke). This alters the absolute net cost of voting but otherwise doesn't change our analysis.

Could you relate this to Bryan Caplan's book "The Myth of the Rational Voter" (who is broadly supportive of your claim)?

My only criticism of this paper is that you ignore risk framing. Which occurs when people have a hard time understanding very low probability events and choose to ignore them, even when they have large payouts.

I discussed Bryan's book here. I think his ideas are consistent with ours, although clearly he's defining "rationality" differently than we do.

Regarding risk framing etc.: I do not believe and would not want to claim that our rationality story is an accurate model of people's actual psychology. Our point is (if you accept our assumptions) that it _is_ rational for many people to vote, so in that sense it's not such a surprise that it happens. Also, occasional close elections such as 2000 remind people that their vote indeed can matter.

This other link might also be interesting to you, where I shoot down someone's published claim that "In decision-theoretic terms it is not rational to vote."

a) you receive $1 million dollars, or
b) everyone in the US receives $10.

I would prefer (a) to (b). I'm fairly sure, though not certain, that if you are reading this, you would also prefer (a) to (b). Are there arguments against this?

If people prefer (a) to (b), then it probably isn't rational for people to vote. In your example above, the $1.5B "lottery ticket" has gone way down in value. What am I missing here?

In the past, Andrew has said that people simply follow different rules when voting than when making other decisions, like the one above. That may be true; people do all sorts of things. However, rationality, while not implying selfishness, should at least require consistency, no? So why would you want different "social preferences" while voting than while making the decision above?

I think you're missing the point of the "rationality argument", because I don't think your estimate for the probability of being decisive is remotely plausible. 1 in ten million chance of being decisive in New York?

If the outcome is uniform over a range including 50%, then sure, the probability of being decisive goes as 1/n. But if we use a much more standard assumption of the outcome being binomial with a mean not exactly 50% Democratic, then the probability of being decisive declines exponentially with n.

There are 7 million votes cast (more or less) in an election in New York, and in New York the mean is something like 60% Democratic. Using standard binomial calculation, the probability of a tied election (and thus my vote being decisive) is about 6*10^(-62055). If the mean were 51% Democratic, the probability of a tied election is only on the order of 10^-612. At 50.1%, it goes down to 10^-10. At exactly 50%, though, we're at .0003 — a good 3/10000 chance of being decisive!

The reason why a standard argument says that voting is irrational *isn't* that my personal benefits are small. It's that the likelihood of being decisive in any election which isn't expected to be completely even is so small that no matter how big my personal benefit is, it won't matter. If voting costs a dollar and I "rationally" decide to vote based on my chance of being decisive, then to vote in New York I'd have to have a benefit of about $ 10^60000 for swinging the vote. I don't care how you value social benefits, the value isn't that high.

I agree with you on the scenario of $1 million vs. $10. It's tricky because once you put it in dollar terms it becomes pretty meaningless: for the simplest case, if you multiply everyone's dollar assets by X, that's pretty much just a case of inflation! What we were talking about was, in some sense, the "equivalent" of the $10 benefit or the $100 benefit or whatever in happiness. You can use a dollar scale to calibrate this but the dollar scale breaks down if you try to consider it too literally. The point is really that we're voting about big issues such as the economy, the war, the environment, abortion, etc., that affect millions of people. If you think the war is a good thing (or a bad thing), it's not just about how it affects you personally.

Alex,

I said "at best" 1 in 10 million in N.Y. I agree that depending on your model you can get the probability down lower. But it's not as low as you say because there is a possibility, a not completely negligible probability, of a vote swing in N.Y. or the mid-Atlantic relative to the rest of the country. It happens. The model you describe is ocmpletely, completely wrong because it assumes the probability is known to be .6, but really this probability is unknown. We discuss this issue extensively in our papers, including the linked paper above and in more detail in our paper on estimating the probability of events that have never occurred: When is your vote decisive? and in these two other papers here, and here.

The short answer is you have to dump the binomial distribution model (which is based on probabilities that are exactly known) and move to a forecasting model that explicitly acknowledges pre-election uncertainty.

What if we replace $10 and $1 million with "magically created goods that would currently be worth $10 / $1 million." For example, replace $10 with an equivalent amount of food, clothing, shelter, etc. of the person's choice. That removes inflation as a factor. Does that make sense?

If so, then does your choice of (a) vs. (b) change? If your choice remains the same, then voting seems inconsistent with that choice. What am I missing?

Yes, in this case I think (or I'd like to think) that I'd swallow hard and not take the $1 million. It's like giving $1M to charity and having it converted into $3 billion, a 3000-to-1 multiplier. Of course, I'd hope that they wouldn't just spend it all on drugs…

Okay. I think I'm more or less convinced — I guess the key is that the uncertainty in the true probability stays pretty much constant as N increases?

Continuing with my New York example with 7,000,000 voters, I used a beta distribution to model the underlying uncertainty of probabilities and a (normal approximation of a) binomial to model vote outcomes given probabilities. I used Mathematica to calculate integrals.

Taking a beta distribution with a standard deviation of about .03, if the mean is .5 then Prob(tied election) is about 2*10^-6. (I used alpha=beta=138).

If the mean is .55 then Prob(tied election) is about 5*10^-7. (alpha=150, beta=123).

If the mean is .6 then Prob (tied election) is about 9*10^-9. (alpha=159, beta=106).

The numbers surprised me (though they sort of make sense, now that I'm thinking about it more) but I guess you already knew this stuff. Thanks for your response.

No name: I can't get a million dollars by voting. I can, however, give everybody in my country ten dollars by voting.

More to the point, nobody ever offered me a million dollars not to vote. I must admit I'd be tempted to skip the polls that year if I was offered such a deal, but it hasn't happened yet. And so instead I go to vote, to get that national ten dollar benefit.

I don't think you can give everyone in your country ten dollars by voting. I think you can get a one-in-ten-million (or whatever) chance of giving everyone in your country ten dollars. If that unlikely outcome isn't worth a million dollars to you (i.e. you would prefer (a) to (b)) then voting wouldn't be rational (again, depending on the specific values, probabilities, etc.) Does this make sense?

The question isn't whether anyone is offering you money to vote or not vote. The answer to the (a) vs. (b) question lets us know about what "social preference" you have (the "alpha" in the linked paper). Once you know that, then you can figure out whether you should vote or not. I think that most people would choose (a) over (b), and therefore are being inconsistent when they vote. If you believe all this, then that inconsistency makes voting irrational (and not the "selfish" vs "social" preference issue).

Andrew (and others):

Maybe I'm way off base, but I think anyone who would choose (b) over (a) would be quite a heroic person (for example, the "I'd like to think" comment in your reply seems to imply that you could see yourself choosing the $1 million). If someone actually did this or anything like this, we would see it on the news. I wouldn't think it is common at all. Am I wrong? Would most people reading this choose to give everyone in the US $10 over getting $1 million for themselves? Would you think that most people in the US would make that choice, if it were given to them. Maybe a poll would be interesting.

If not, the argument still goes through: if most people would choose (a) over (b), then voting is inconsistent with the social preference implied by that choice. Do you disagree?

I should clarify that these probabilities are based on forecasts a couple months before the election. If you estimate the probabilities on the morning of election day, you could get much lower probabilities for New York in recent presidential elections. I'm formulating the voting decision as something that happens gradually over the entire campaign or even one's entier life, as you consider whether to make voting a regular activity.

No name,

If I were offered the opportunity to determine the election winner all by myself, I'd certainly take that over the $1 million, with no hesitation at all. Also, if I could either have $1000 or a 1/1000 probability of determining the winner, I'd again certainly take the 1/1000 probability of determining the winner. One difficulty of your formulation is the framing as $10 for each person. The benefits of the preferred candidate winning can't really easily be converted to money. In some abstract sense, issues such as war, abortion, health, etc., can be thought of as equivalent to some amount of utility implied by money, but it's not always so clear how to do so.

Regarding your other point, lots of people give hundreds or thousands of dollars to charity with no matching funds at all. Here you're talking about a 3000-to-1 ratio of matching funds, which is pretty good. On the other hand, your hypothetical charity is not targeted–it's just $10 to everybody–which complicates things a bit. If I think that my preferred candidate can improve the economy, this is lots more than $10 each.

The $1 million for you vs $10 for everyone (a vs. b) question is to calculate your "alpha" which represents your social preferences. Neither the $1 million nor the $10 each should be compared to the actual benefits of someone winning the election. We can answer the (a) vs. (b) question quite apart from any thoughts about voting. For example, the fact that you would rather choose the next president than get $1 million is not relevant to any point of my argument. Do you disagree?

The idea is that, since rationality implies consistency, the "social preferences" you demonstrate in my artificial (a) vs (b) choice should apply to the voting decision.

In your post, you used an example of $50 per person ("not implausible") and a 1 in ten million chance of affecting the outcome. You then argued that that represented a lottery winning of $1.5 billion. If you choose (a) over (b), then that lottery ticket is much lower value, by a factor of at least 3000 to 1. So it looks more like $1.5B/3000 = $500K lottery ticket. One in ten million of that is $0.05, which implies to me that voting would be irrational, as the process of voting is more costly that that. Changing the numbers we both assumed will change the outcome; then the arguments would need to become more substantive (e.g. someone might argue that a particular candidate would give a huge improvement, whereas others might argue that "all politicians are equally bad", and others might say that there is much more uncertainty in the value provided that you can't really tell the difference, etc.). What I'm saying is that, using the numbers you used in your blog post and the arguments you used in your paper, voting would still be irrational. What is your reaction to that? Am I mistaken?

You have argued that you would bite the bullet and choose (b) over (a). I think that most people would not. I don't know though.

Instead of "matching funds", you may want to think about "bang-for-the-buck." For example, I might choose (a) over (b), but still give to a charity that works on child health in developing countries. The reason is that for only a few dollars, I could save a child's life, much more than a mere 3000-to-1 payoff however you measure it. So even though there are no matching funds, giving to (high quality) charities have a much higher bang-for-the-buck than choosing (b) over (a). Therefore, the fact that people give to charity does not affect the argument that voting is irrational. Comments?

As for the difficulty in assessing dollar values for intangibles, I take your point. So is your new argument for the rationality of voting that the outcomes simply can't be measured well? That seems very different from what you said in the blog post, no?

First, thanks a lot for putting in the time to express your argument in writing. I appreciate the dialogue.

That said, I still stand by our article. You make a good point that the "alpha" parameter is not precisely defined. The point about using money as a scale is that it is a shorthand for a utility measure that one would apply for one's own money. As I said, I think it's perfectly rational to prefer the 1/1000 chance of affecting the election outcome to a certain $1000, and this could be sliced other ways, as discussed in the article. As the blog exchange has brought out, it's tricky to try to "cash out" benefits to others, even hypothetically, but, no, I don't think that makes it irrational to vote. It's also hard to cash out health gains, but I don't think it's irrational to spend money on health care (or, for that matter, to vote for policies that you think could improve others' health).

The reasoning in our paper is approximate, as is all formal theory in social science, but I think the social benefit term does add a lot to the model. It makes a lot more sense than the previous model, which was that people vote either because of being misinformed (overestimating small probabilities) or as a consumption good. Both these traditional explanations have some truth to them, but they also miss a lot (as discussed in the paper), not explaining phenomena such as when people vote, why they say they vote, and why the contribute small amounts to political campaigns.

You make solid utilitarian points, given your $50/person assumption. But I wonder how confident anyone has a right to be about their political preferences. Would you really put the expected value of having your chosen candidate win, instead of the candidate he'd displace, at 1.5 billion? For Hanson-style reasons, I'm nowhere near totally confident I know who the right candidate even is, let alone how much better he is than the other guy.

For someone who isn't confident who the right candidate is, it may not be rational for them to vote. But if someone believes that their preferred candidate is equivalent to a 1.5 billion lottery ticket, then it is rational for that person to vote conditional on their beliefs.

The only caveat is how rational are your beliefs that your preferred candidate is so much better than the other. The argument that it can be rational to vote seems to take advantage of a typical overconfidence bias… whether or not this really matters for the argument of rationality, I'm not sure. But rationality based off biased beliefs doesn't seem quite right.

A 1/10,000,000 chance of giving $50 worth of goods to each American is really only an expected gain of 5/1000 of one cent to each American. Only a theorist would think that most people would value such a small gain, so broadly distributed, over an hour of their own time. I certainly would not spend an hour to give a paperclip to every man, woman and child in America. Would any of you?

Yes, youall are probably right. At some level, the whole idea of rationality falls apart somewhere. For example, suppose you vote based on candidates' positions on abortion. This is a reasonable rationale for voting but there's no really good way of framing it so that it's "rational" for some people to support legal abortion and "rational" for others to oppose it. These are just different preferences (and different value systems).

In any case, I was responding to the argument that says: It's irrational to vote because your probability of making a difference is so small, it's worse than buying a lottery ticket, etc. If, for whatever reason, you care about which candidate wins and you think he or she could really make the world much better (or that the opposition could make the world much worse), then voting can be perfectly rational without any need to bring in the "consumption value" of voting, "civic duty," etc.